If z is a complex number, then minimum value of
(i) `| z| + | z - 1| + | 2 z - 3| and `
(ii) `| z + 1| + | z - 1|` is

Answer

To solve the problem, we will analyze both parts step by step.

### Part (i): Find the minimum value of \( |z| + |z - 1| + |2z - 3| \)

1. **Understanding the terms**:
- \( |z| \) represents the distance of the complex number \( z \) from the origin (0,0).
- \( |z - 1| \) represents the distance of \( z \) from the point (1,0).
- \( |2z - 3| \) can be rewritten as \( |2(z - \frac{3}{2})| = 2|z - \frac{3}{2}| \), which represents twice the distance of \( z \) from the point \( (\frac{3}{2}, 0) \).

2. **Using the triangle inequality**:
- We can apply the triangle inequality which states that \( |a + b| \leq |a| + |b| \).
- We can rearrange the terms:
\[ |z| + |z - 1| + |2z - 3| \geq |z + (z - 1) + (2z - 3)| \]
- Simplifying the expression inside the absolute value gives:
\[ |z + z - 1 + 2z - 3| = |4z - 4| = 4|z - 1| \]

3. **Finding the minimum value**:
- The minimum value of \( |z - 1| \) occurs when \( z = 1 \), which gives \( |z - 1| = 0 \).
- Thus, substituting \( z = 1 \) into our inequality:
\[ |z| + |z - 1| + |2z - 3| \geq 4 \cdot 0 = 0 \]
- However, we need to evaluate the original expression at \( z = 1 \):
\[ |1| + |1 - 1| + |2 \cdot 1 - 3| = 1 + 0 + 1 = 2 \]

4. **Conclusion for part (i)**:
- Therefore, the minimum value of \( |z| + |z - 1| + |2z - 3| \) is **2**.

### Part (ii): Find the minimum value of \( |z + 1| + |z - 1| \)

1. **Understanding the terms**:
- \( |z + 1| \) represents the distance of \( z \) from the point (-1,0).
- \( |z - 1| \) represents the distance of \( z \) from the point (1,0).

2. **Using the triangle inequality**:
- We can apply the triangle inequality again:
\[ |z + 1| + |z - 1| \geq |(z + 1) + (z - 1)| = |2z| = 2|z| \]

3. **Finding the minimum value**:
- The minimum value of \( |z| \) occurs when \( z = 0 \), which gives \( |z| = 0 \).
- Thus, substituting \( z = 0 \) into our inequality:
\[ |0 + 1| + |0 - 1| = |1| + |-1| = 1 + 1 = 2 \]

4. **Conclusion for part (ii)**:
- Therefore, the minimum value of \( |z + 1| + |z - 1| \) is also **2**.

### Final Answers:
- (i) Minimum value of \( |z| + |z - 1| + |2z - 3| \) is **2**.
- (ii) Minimum value of \( |z + 1| + |z - 1| \) is **2**.